A Note on Terai's Conjecture Concerning the Exponential Diophantine Equation x^{2}+b^{y}=c^{z}

Refik Keskin; Zafer Şiar

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arXiv · 2105.14814

A Note on Terai's Conjecture Concerning the Exponential Diophantine Equation x^{2}+b^{y}=c^{z}

Refik Keskin·Zafer Şiar

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Abstract

Let (a,b,c) be a primitive Pythagorean triple, i.e., a^{2}+b^{2}=c^{2} with gcd(a,b,c)=1, a even and b odd. Terai's conjecture says that the Diophantine equation x^{2}+b^{y}=c^{z} has only the positive integer solutions (x,y,z)=(a,2,2). In this study, we prove that Terai's conjecture is true when b is a product of two primes and c=5(mod8).

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Refik Keskin, Zafer Şiar. 2021-05-31. A Note on Terai's Conjecture Concerning the Exponential Diophantine Equation x^{2}+b^{y}=c^{z}. https://arxiv.org/abs/2105.14814

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